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June  2021, 20(6): 2187-2209. doi: 10.3934/cpaa.2021063

Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity

Andrew Comech 1, and  Elena Kopylova 2,,

1. 

Texas A & M University, College Station, TX, Institute for Information Transmission Problems, Moscow, Russia
2. 

Vienna University, Vienna, Austria, Institute for Information Transmission Problems, Moscow, Russia

* Corresponding author

Received  August 2020 Revised  February 2021 Published  June 2021 Early access  April 2021

Fund Project: The second author is supported by the Austrian Science Fund (FWF), Grant P 34177-N

We obtain explicit characterization of orbital and spectral stability of solitary wave solutions to the $ {\bf{U}}(1) $-invariant 1D Klein–Gordon equation coupled to an anharmonic oscillator. We also give the complete analysis of the spectrum of the linearization at a solitary wave.

Keywords: Klein-Gordon equation, concentrated nonlinearity, orbital stability, spectral stability, solitary waves.
Mathematics Subject Classification: Primary: 35L70; Secondary: 47F05.
Citation: Andrew Comech, Elena Kopylova. Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2187-2209. doi: 10.3934/cpaa.2021063
References:
[1]

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2005. doi: 10.1007/978-3-642-88201-2.  Google Scholar

[2]

N. Boussaïd and A. Comech, Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schrödinger operators, preprint, arXiv: 2101.11979. Google Scholar

[3]

V. Buslaev, A. Komech, E. Kopylova and D. Stuart, On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator, Commun. Partial Differ. Equ., 33 (2008), 669-705.  doi: 10.1080/03605300801970937.  Google Scholar

[4]

E. Csobo, F. Genoud, M. Ohta and and J. Royer, Stability of standing waves for a nonlinear Klein–Gordon equation with delta potentials, J. Differ. Equ., 268 (2019), 353-388.  doi: 10.1016/j.jde.2019.08.015.  Google Scholar

[5]

A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Commun. Pure Appl. Math., 56 (2003), 1565-1607.  doi: 10.1002/cpa.10104.  Google Scholar

[6]

M. Grillakis, J. Shata and W. Strauss, Stability theory of solitary waves in the presence of symmetry, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[7]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.   Google Scholar

[8]

A. Komech and A. Komech, Global attractor for a nonlinear oscillator coupled to the Klein–Gordon field, Arch. Ration. Mech. Anal., 185 (2007), 105-142.  doi: 10.1007/s00205-006-0039-z.  Google Scholar

[9]

A. Komech and A. Komech, Global attraction to solitary waves for Klein–Gordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 855-868.  doi: 10.1016/j.anihpc.2008.03.005.  Google Scholar

[10]

A. Komech, E. Kopylova and D. Stuart, On asymptotic stability of solitons in a nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 11 (2012), 1063-1079.  doi: 10.3934/cpaa.2012.11.1063.  Google Scholar

[11]

A. Kolokolov, Stability of the dominant mode of the nonlinear wave equation in a cubic medium, J. Appl. Mech. Tech. Phys., 14 (1973), 426-428.  doi: 10.1016/0021-8928(74)90131-2.  Google Scholar

[12]

E. Kopylova, On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 3031-3046.  doi: 10.1016/j.na.2009.01.188.  Google Scholar

[13]

E. Kopylova, On asymptotic stability of solitary waves in discrete Klein–Gordon equation coupled to a nonlinear oscillator, Appl. Anal., 89 (2010), 1467-1492.  doi: 10.1080/00036810903277176.  Google Scholar

[14]

M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), 10-56.  doi: 10.1016/0022-1236(82)90084-2.  Google Scholar

[15]

M. Ohta and G. Todorova, Strong instability of standing waves for the nonlinear Klein–Gordon equation and the Klein–Gordon–Zakharov system, SIAM J. Math. Anal., 38 (2007), 1912-1931.  doi: 10.1137/050643015.  Google Scholar

show all references

References:
[1]

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2005. doi: 10.1007/978-3-642-88201-2.  Google Scholar

[2]

N. Boussaïd and A. Comech, Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schrödinger operators, preprint, arXiv: 2101.11979. Google Scholar

[3]

V. Buslaev, A. Komech, E. Kopylova and D. Stuart, On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator, Commun. Partial Differ. Equ., 33 (2008), 669-705.  doi: 10.1080/03605300801970937.  Google Scholar

[4]

E. Csobo, F. Genoud, M. Ohta and and J. Royer, Stability of standing waves for a nonlinear Klein–Gordon equation with delta potentials, J. Differ. Equ., 268 (2019), 353-388.  doi: 10.1016/j.jde.2019.08.015.  Google Scholar

[5]

A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Commun. Pure Appl. Math., 56 (2003), 1565-1607.  doi: 10.1002/cpa.10104.  Google Scholar

[6]

M. Grillakis, J. Shata and W. Strauss, Stability theory of solitary waves in the presence of symmetry, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[7]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.   Google Scholar

[8]

A. Komech and A. Komech, Global attractor for a nonlinear oscillator coupled to the Klein–Gordon field, Arch. Ration. Mech. Anal., 185 (2007), 105-142.  doi: 10.1007/s00205-006-0039-z.  Google Scholar

[9]

A. Komech and A. Komech, Global attraction to solitary waves for Klein–Gordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 855-868.  doi: 10.1016/j.anihpc.2008.03.005.  Google Scholar

[10]

A. Komech, E. Kopylova and D. Stuart, On asymptotic stability of solitons in a nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 11 (2012), 1063-1079.  doi: 10.3934/cpaa.2012.11.1063.  Google Scholar

[11]

A. Kolokolov, Stability of the dominant mode of the nonlinear wave equation in a cubic medium, J. Appl. Mech. Tech. Phys., 14 (1973), 426-428.  doi: 10.1016/0021-8928(74)90131-2.  Google Scholar

[12]

E. Kopylova, On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 3031-3046.  doi: 10.1016/j.na.2009.01.188.  Google Scholar

[13]

E. Kopylova, On asymptotic stability of solitary waves in discrete Klein–Gordon equation coupled to a nonlinear oscillator, Appl. Anal., 89 (2010), 1467-1492.  doi: 10.1080/00036810903277176.  Google Scholar

[14]

M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), 10-56.  doi: 10.1016/0022-1236(82)90084-2.  Google Scholar

[15]

M. Ohta and G. Todorova, Strong instability of standing waves for the nonlinear Klein–Gordon equation and the Klein–Gordon–Zakharov system, SIAM J. Math. Anal., 38 (2007), 1912-1931.  doi: 10.1137/050643015.  Google Scholar

Figure 1.  The relation between $ \lambda\in\sigma(H_\kappa(\omega)) $ and $ \Lambda\in\sigma(L_\kappa(\omega)) $ for $ \omega\in(-m,m)\setminus\{0\} $, $ \kappa>0 $
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Figure 2.  Location of simple nonzero eigenvalues $ \pm\lambda $
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